Math 151 Handout Example: Using the Definition of the ...



Math 151 Handout Examples: Using the Definition of the Derivative,

Example 1: Use the definition of the derivative to find [pic] if [pic].

Solution: Using the limit definition of the derivative, we see that

[pic]

Example 2: Find the equation of the line tangent to the graph of [pic] at the point (0, -1).

Solution: To find the equation of any line, including a tangent line, we need to know the line’s slope and a point on the line. Since we already have a point on line, we must find the tangent line’s slope, which is found using the derivative. Using the limit definition of the derivative, we see that

[pic]

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Using [pic], we can now find the slope at the give point (0, -1).

[pic].

Using [pic], we see that from the slope intercept equation [pic]

that

[pic]

To find b, use the fact that at the point (0, -1), [pic] and [pic]. Thus

[pic]

giving [pic]. Thus, the equation of the tangent line is:

[pic].

[pic]

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